Isolated horizons of the Petrov type D

Transcription

1 Isolated horizons of the Petrov type D 1, 2). Denis Dobkowski-Ryłko, Jerzy Lewandowski, Tomasz Pawłowski (2018); 3). JL, Adam Szereszewski (2018); 4). DDR, Wojtek Kamiński, JL, AS (2018); Uniwersytet Warszawski

2 Null surface stationary to the 2nd order 3d null surface in 4d spacetime Exists: t M t H t = ` Such that L t g µ =0 [L t, r µ ]=0 L t R µ =0 } ` `µ`µ = 0 Assumption about M : G µ + g µ =0 2

3 Intrinsic geometry g µ, r µ - spacetime geometry and derivative µ, M a, b H H r a`b =! a`b apple` =! a`a - rotation potential - surface gravity g ab, r a Assumption: apple` 6= 0 G µ + g µ =0 ) ` r a apple` = L`! a =0 - zeroth law of thermodynamics (g ab,! a ) determine r a determine g µ, r µ, R µ 3

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5 Data on a 2d slice: summary 2d-surface S Endowed with (g AB,! A ) modulo! 0 A =! A + f,a r A - the corresponding derivative r A g BC =0 r [A r B] f =0 Scalar invariants: Gaussian curvature: K d! =: O darea combined: := 1 2 (K + io) 5

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sha1_base64="c0hy6vyshcmcjwecsxh5+gczxqi=">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</latexit> The Weyl tensor in Newman-Penrose formalism - Spacetime Weyl tensor in the null frame formalism may be expressed by the following complex valued N-P components: 0 = C 4141, 1 = C = C 4123, 3 = C 3432, 4 = C Four components are constant along the null generators of H: D I =0, I =0, 1, 2, 3 - Additionally we assume: D 4 =0 <latexit 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sha1_base64="vlmqyc3oac/un4tgsrfogxjdpve=">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</latexit> - The components and vanish due to vanishing of the expansion and shear of : - is related to the complex invariant: 2 0 <latexit sha1_base64="oi83ggbsi8cgvio9xbqfh5krswg=">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</latexit> 1 0 = 1 =0 2 = + 6 6

7 Possible Petrov types The spacetime Weyl tensor at data H is determined by the (S, g AB,! A ) Theorem: The possible Petrov types of H are: I, II, D, III, N, O wherein: + 6 =0, O, K = 3 d! = =0 ) generically II, unless 7

8 The Petrov type D equation We use a null 2-frame g AB = m A m B + m A m B darea BC = i( m B m C m C m B ) Theorem: At H the spacetime Weyl tensor is of the Petrov type D iff the following two conditions are satisfied: + 6 6=0 m A m B r A r B ( + 6 ) 1 3 =0 8

9 In local conformally flat coordinates g AB dx A dx B = 2 P 2 dzd z m A = P@ z The Petrov type D z (P z ( + 6 ) 1 3 )=0 9

10 The Petrov type D equation as integrability condition for the near horizon geometry equation Theorem: Suppose (g AB,! A ) satisfy the NHG equation, namely r (A! B) +! A! B ( K)g AB =0 Then they also satisfy the Petrov type D equation: m A m B r A r B ( ) 1 3 =0 10

11 Non-twisting of the second principal null direction of the Weyl tensor Theorem: Suppose (g AB,! A ) satisfy the NHG equation, namely r (A! B) +! A! B ( K)g AB =0 Then the null vector n 0 orthogonal to the corresponding slice S S! A g AB is a double principal direction of the spacetime Weyl n 0 tensor at H. ` In that case there exists another symmetry t that is extremal 11

12 <latexit sha1_base64="h0na9nf4j8rqkdvwkkf0jqqmasu=">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</latexit> <latexit 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sha1_base64="xyx4dsnzvg1xka9txwbh927i4/e=">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</latexit> <latexit sha1_base64="wnxoiltev+k4tjhluk6i/p85yu4=">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</latexit> No-hair theorem for axisymmetric solutions to the Petrov type D equation. Theorem The family of axisymmetric solutions of the Petrov type D equation with (or without) cosmological constant defined on a topological sphere can be parametrized by two numbers (A,J): the area and angular momentum, respectively. They can take the following values: for > 0:J 2 1, 1 for A 2 0, 12 and J 2 apple 0, r A A for A 2 12, 1 for < 0:J 2 1, 1 and A 2 0, 1 Lewandowski, Pawlowski (2003) for =0 12

13 Embeddability of the axisymmetric solutions Every solution defines a type D isolated horizon whose intrinsic geometry coincides with the intrinsic geometry of a non-extremal Killing horizon contained in one of the following spacetimes: 1). Kerr - (anti) de Sitter; 2). Schwarzschild - (anti) de Sitter ; 3). Near horizon limit spacetime near an extremal horizon contained either in the K(a)dS or S(a)dS spacetime;

14 <latexit sha1_base64="p28k2bitvmhrn0dznr3byj55+xo=">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</latexit> <latexit sha1_base64="p28k2bitvmhrn0dznr3byj55+xo=">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</latexit> <latexit sha1_base64="p28k2bitvmhrn0dznr3byj55+xo=">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</latexit> <latexit sha1_base64="cv0a4vv/ik807b2xd8lwqbz3n6g=">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</latexit> The Petrov type D equation on S of genus > 0 g AB dx A dx B = 2 P 2 dzd z The Petrov type D equation: m A = P@ z(p z( + 6 ) 1 3 )=0 z = F (z) P 2 ) F (z)@ z is a globally defined holomorphic vector field ) F (z) = const F (z) =0 if genus =1 if genus >1 14

15 The Petrov type D equation on S of genus > 0 Theorem. Suppose S is a compact 2-surface of genus >0. The only solutions to the Petrov type D equation with a cosmological constant are (g,!) such that d! =0 and K = const 6= 3 Remark: There are no rotating solutions 15

16 <latexit sha1_base64="1sfqwmntraxy5xrpaidjtnvduqm=">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</latexit> <latexit sha1_base64="1sfqwmntraxy5xrpaidjtnvduqm=">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</latexit> <latexit sha1_base64="1sfqwmntraxy5xrpaidjtnvduqm=">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</latexit> <latexit sha1_base64="uxhjh2eaj3siguuswec1ipbtx9q=">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</latexit> A bifurcated Petrov type D horizon: data H 0 H S g 0 AB = g AB! 0 A =! A M. J. Cole, I. Racz, J. A. Valiente Kroon, 2018 J. Lewandowski, A. Szereszewski, = 16

17 A bifurcated Petrov type D horizon: equations The Petrov type D equations for H: m A m B r A r B ( ) 1 3 =0 and for H : m A m B r A r B ( ) 1 3 =0 hold simultaneously on S 17

18 In local conformally flat coordinates g AB dx A dx B = z (P z ( + 6 ) 1 3 )=0 P 2 dzd z ) m A = P@ z ( + 6 ) 1 3 = F (z) P z (P z ( + 6 ) 1 3 ( + 6 ) 1 Ḡ( z) ) 3 = z F (z) P 2 = P z Ḡ( z) P 2 ) L g AB =0 :=F (z)@ z Ḡ( z)@ z ) L d! =0 18

19 The axial symmetry without the rigidity theorem Theorem: Suppose (g AB,! A ) defined on S satisfy the Petrov type D equation m A m B r A r B ( ) 1 3 =0 and the conjugate one m A m B r A r B ( ) 1 3 =0 Then, there is a vector field at S such that L g AB =0 and L d! =0 A =Re/Im darea A ( + 6 )

20 <latexit sha1_base64="gjyfrwhu/0jiirx6ldlmtggc32y=">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</latexit> <latexit sha1_base64="gjyfrwhu/0jiirx6ldlmtggc32y=">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</latexit> <latexit sha1_base64="gjyfrwhu/0jiirx6ldlmtggc32y=">aaadrhicjvfdsxwxfl0z1dbpdlvrky/brvbklxlb2r4ifrwifmqhq4kjsyab3qyzh2qybrkg/51/ofgdchv8e1/fm2ywaku0w8yce+49j7m5cs5foypgzpofjyw+fze2pje5nf3yvep1m50ikxxjbzbjto3ftobsplythzz8l1ecjrhku/hrhsnv/ukqefn6qx/n/cch/vt0bkmaqu7jjiqpisnhgnfgnuqpjsxtra0bmrho9xfj+6inkkvculquydadujaqylrw3+qavcodrpqvj9gldfwphjgshvb/nb+wcouenuyzaav2kfsgdkajbm1njd8qqrcyyfbcahxs0iglucjw2ycqasiro4akoyvi2dyhgizqw2ivxwqk7bf++xjtozbf2hgwvs1wf4mvqiwbbdrkwkcqm92izzfw2bapevfw05ztgp+x80qq1fat2cd0w8qn6kwvgnrwxfygskfcmqy75lxkeyvm5orwvxodcuqm7mjeiwzwobxnyjwf7d3clbx5v7bssczmrraec3nkhhd4/zjvg53lvhi0wu8fm6vrbtrjmafzsijz/ayrsanb0ebvp96k99ab9u/9c//svxqu+p7tzmcd5v/fab5ovqe=</latexit> <latexit sha1_base64="oh2oixzusy/jplao2qkr1v6urwa=">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</latexit> Summary The type D equation: m A m B r A r B 1 2 K 1 2 io =0 Non-twisting of the second double principal vector if: r (A! B) +! A! B ( K)g AB =0 All the axisymmetric solutions of the type D eq. on topological sphere parametrized by (A, J); All solutions on genus>0 derived (non-rotating); The extra (axial) symmetry in the case of bifurcated horizon; Open problems: existence of non-axisymmetric solutions on topological sphere

21 Thank You